1 Introduction
Often the first step in the analysis of a spatial point pattern is to estimate its intensity function. Various nonparametric estimators are available to do so. Some techniques are based on local neighbourhoods of a point, expressed for example by its nearest neighbours [7], its Voronoi [11] or Delaunay tessellation [13, 14]. By far the most popular technique, however, is kernel smoothing [6]. Specifically, let be a point process that is observed in a bounded open subset of
and assume that its first order moment measure exists as a
finite Borel measure and is absolutely continuous with respect to Lebesgue measure with a Radon–Nikodym derivative known as its intensity function. A kernel estimator of based on takes the form(1) 
The function is supposed to be kernel, that is, a
dimensional symmetric probability density function
[15, p. 13]. The choice of bandwidth determines the amount of smoothing. In principle, the support of as a function of could overlap the complement of . Therefore, various edge corrections have been proposed [2, 9]. In the sequel, though, we will be concerned with very small bandwidths, so this aspect may be ignored.The aim of this paper is to derive asymptotic expansions for the bias and variance of (
1) in terms of the bandwidth. This problem is well known when dealing with probability density functions. Indeed, there exists a vast literature, for example the textbooks [3, 15, 16] and the references therein. In a spatial context, bandwidth selection is dominated by ad hoc [2] and nonparametric methods [5]. The first rigorous study into bandwidth selection to the best of our knowledge is that by Lo [10] who studies infill asymptotics for spatial patterns consisting of independent and identically distributed points. Our goal is to extend his approach to point processes that may exhibit interactions between their points and to investigate adaptive versions thereof.The plan of this paper is as follows. In Section 2 we focus on the regime in which independent copies of the same point process are superposed and the bandwidth tends to zero as tends to infinity but does not depend on the points of the pattern. We derive Taylor expansions and deduce the asymptotically optimal bandwidth. Intuitively, however, one feels that in sparse regions more smoothing is necessary then in regions that are rich in points. Indeed, in the context of estimating a probability density function, Abramson [1] proposed to scale the bandwidth in proportion to the square root of the density. Analogously, in Section 3 we let decrease in proportion to the square root of the intensity function and show that by doing so the bias can be reduced. For the sake of readability, all proofs are deferred to Section 4.
2 Infill asymptotics
Let be independent and identically distributed point processes for which the first order moment measure exists, is locally finite and admits an intensity function . For , let
denote the union. Upon taking the limit for , one obtains an asymptotic regime known as ‘infill asymptotics’ [12]. Since the are independent, the intensity function of is . Therefore , , may be estimated by
(2) 
Lemma 1
Let be a point process observed in a bounded open subset whose factorial moment measures exist up to second order and are absolutely continuous with intensity function and second order product densities . Let be a kernel. Then the first two moments of (1) are
and
The proof follows directly from the definition of product densities, see for example [4, Section 4.3.3]. Provided , the variance of can expressed in terms of the pair correlation function defined by as
For Poisson processes, the first integral vanishes as .
In this paper, we will restrict ourselves to kernels that belong to the Beta class
(3) 
for . Here is the closed unit ball in centred at the origin. The normalising constant will be abbreviated by
(4) 
Note that Beta kernels are supported on the compact unit ball and that their smoothness is governed by the parameter . Indeed, the box kernel defined by is constant and therefore continuous on the interior of the unit ball; the Epanechnikov kernel corresponding to the choice is Lipschitz continuous. For the function is times continuously differentiable on .
The following Lemma collects further basic properties of the Beta kernels. The proof can be found in Section 4.1.
Lemma 2
For the Beta kernels , , defined in equation (3), the integrals
vanish for all such that . Furthermore
is finite and so are, for all ,
as well as, for and ,
Their values do not depend on the particular choices of and .
For the important special case ,
Proposition 1
Let be independent and identically distributed point processes observed in a bounded open subset . Assume that their factorial moment measures exist up to second order and are absolutely continuous with strictly positive intensity function and second order product densities . Write for the union, , and let be a Beta kernel (3) with . Then the mean squared error of (2) is given by
The first term in the above expression is the squared bias. It depends on and the bandwidth but not on . The remaining terms come from the variance and depend on , on , on and on .
Our aim in the remainder of this section is to derive an asymptotic expansion of the mean squared error for bandwidths that depend on in such a way that as . In order to achieve this, first recall some basic facts from analysis. Let be an open subset of and denote by the class of functions for which all order partial derivatives exist and are continuous on . For such functions the order of taking partial derivatives may be interchanged and the Taylor theorem states that if and for all , then a can be found such that
(5) 
where is the tuple and
for .
We are now ready to state the main result of this section, generalising [10, Theorem 2] for the union of independent random points. The proof can be found in Section 4.2.
Theorem 1
Let be i.i.d. point processes observed in a bounded open subset with welldefined intensity function and pair correlation function . Suppose that is bounded and that is twice continuously differentiable with second order partial derivatives , , that are Hőlder continuous with index on , that is, there exists some such that for all :
Consider the estimator based on the unions , , and Beta kernel , , with bandwidth chosen in such a way that, as , and . Then, for , as ,

.

.
The bias depends on the second order partial derivatives of the unknown intensity function and on the smoothness parameter . The smoothness of the kernel, measured by , also plays a role. The leading term of the variance depends on and on the smoothness of the kernel.
Corollary 1
In words, is of the order . Clearly tends to zero as . Moreover, is of the order to the and therefore tends to infinity with . For the special case ,
Proposition 2
Let be i.i.d. point processes observed in a bounded open subset with welldefined intensity function and pair correlation function . Suppose that is bounded and that is twice continuously differentiable with second order partial derivatives , , that are Hőlder continuous with index on . Consider based on the unions , , and Beta kernel , , with bandwidth chosen in such a way that as , and . Then, for , as ,
3 Adaptive infill asymptotics
Up to now, estimators based on (1) were considered in which the same bandwidth was applied at every point . However, at least intuitively, it seems clear that the bandwidth should be smaller in regions with many points, larger when points are scarce. This suggests that should be decreasing in .
Motivated by similar considerations in the context of density estimation, Abramson [1] suggested to consider pointdependent bandwidths of the form for equal to the square root of the probability density function. He found that a significant reduction in bias could be obtained by the use of such adaptive bandwidths. Our aim in this section is to show that a similar result holds for spatial intensity function estimation.
Define an estimator
of , , that is the average of dataadaptive estimators
(6) 
As in Section 2, is a kernel and the , , are independent and identically distributed point processes on observed in a bounded nonempty open subset for which the first order moment measure exists and admits an intensity function ; is assumed to be a measurable positivevalued weight function on .
The next result summarises the first two moments.
Lemma 3
Let be a point process observed in a bounded open subset , whose factorial moment measures exist up to second order and are absolutely continuous with intensity function and second order product densities . Let be a kernel. Then the first two moments of (6) are
and
The proof follows directly from the definition of product densities, see for example [4, Section 4.3.3]. For the special case , we retrieve Lemma 1.
Provided , the variance of , the average of the , can be expressed in terms of the pair correlation function as
(7) 
We are now ready to state the first main result of this section in analogy to [1, Theorem, p. 1218]. The proof can be found in Section 4.3.
Theorem 2
Let be i.i.d. point processes observed in a bounded open subset with welldefined intensity function and pair correlation function . Suppose that is bounded and that is bounded, bounded away from zero and twice continuously differentiable on with bounded second order partial derivatives , .
Consider the estimator with
based on the unions , , and Beta kernel , , with bandwidth chosen in such a way that, as , and . Then, for , as ,

.

.
In comparison with Theorem 1, the variance is the same as that for a nonadaptive bandwidth. The bias term on the other hand is of a smaller order. Note that, since the leading bias term is not specified, Theorem 2 cannot be used to calculate an asymptotically optimal bandwidth. To remedy this, stronger smoothness assumptions seem needed.
Theorem 3
Let be i.i.d. point processes observed in a bounded open subset with welldefined intensity function and pair correlation function . Suppose that is bounded and that is bounded, bounded away from zero and five times continuously differentiable on with bounded partial derivatives.
Consider the estimator with
based on the unions , , and Beta kernel , , with bandwidth chosen in such a way that, as , and . Then, for , as ,

, where
and .

.
For the important special cases , the expression for may be simplified. All the proofs are given in Section 4.3.
Proposition 3
Consider the framework of Theorem 3 in one dimension . Then the coefficient of in the expansion of is
where and the superscript indicates the fourth order derivative.
Proposition 4
Consider the framework of Theorem 3 in two dimensions . Then the coefficient of in the expansion of is
with , and constants
and
Theorem 3 immediately yields the asymptotically optimal bandwidth, which should be compared with that in Corollary 1.
Corollary 2
The optimal bandwidth and the weights depend on the unknown intensity function. In practice, a nonparametric pilot estimator (for example the one proposed in [5]) would be plugged in.
To conclude this section, we present the analogue of Proposition 2. The proof can be found in Section 4.3.
Proposition 5
Let be i.i.d. point processes observed in a bounded open subset with welldefined intensity function and pair correlation function . Suppose that is bounded and that is bounded, bounded away from zero and five times continuously differentiable on with bounded partial derivatives. Consider with based on the unions , , and Beta kernel , , with bandwidth chosen in such a way that as , and . Then, for , as ,
where is as defined in Theorem 3.
4 Proofs and technicalities
4.1 Auxiliary lemmas for the Beta kernel
Proof of Lemma 2: The first two claims follow from the symmetry of the Beta kernel. Furthermore
Due to the symmetry of the Beta kernel it is clear that the definitions of , and do not depend on the choices of and . First consider the case . By the symmetry of and a change of variables , , it follows that
Similarly,
For dimensions , write and as a repeated integral and note that the innermost integral takes the form
for . By the symmetry and a change of parameters , it follows that
and
in accordance with the claim.
Finally for , can be written as
The inner integral is equal to so
in accordance with the claim.
In the sequel, the following additional properties of the Beta kernels will be needed.
Lemma 4
Consider the Beta kernels with defined in equation (3). Then, for all ,
the integrals of second order products in with respect to vanish and for distinct ,
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